Abstract

We propose a reconstruction scheme for hard x-ray inline holography, a variant of propagation imaging, which is compatible with imaging conditions of partial (spatial) coherence. This is a relevant extension of current full-field phase contrast imaging, which requires full coherence. By the ability to reconstruct the coherent modes of the illumination (probe), as demonstrated here, the requirements of coherence filtering could be relaxed in many experimentally relevant settings. The proposed scheme is built on the mixed-state approach introduced in [Nature 494, 68 (2013)], combined with multi-plane detection of extended wavefields [Opt. Commun. 199, 65 (2001), Opt. Express 22, 16571 (2014)]. Notably, the diversity necessary for the reconstruction is generated by acquiring measurements at different defocus positions of the detector. We show that we can recover the coherent mode structure and occupancy numbers of the partial coherent probe. Practically relevant quantities as the transversal coherence length can be computed from the reconstruction in a straightforward way.

© 2017 Optical Society of America

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References

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  27. M. Krenkel, A. Markus, M. Bartels, C. Dullin, F. Alves, and T. Salditt, “Phase-contrast zoom tomography reveals precise locations of macrophages in mouse lungs,” Sci. Rep. 5, 09973 (2015).
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2017 (1)

J. Hagemann, A.-L. Robisch, M. Osterhoff, and T. Salditt, “Probe reconstruction for holographic X-ray imaging,” J. Synchrotron Rad. 24, 498–505 (2017).
[Crossref]

2016 (4)

P. Li, T. Edo, D. Batey, J. Rodenburg, and A. Maiden, “Breaking ambiguities in mixed state ptychography,” Opt. Express 24, 9038–9052 (2016).
[Crossref] [PubMed]

M. Töpperwien, M. Krenkel, F. Quade, and T. Salditt, “Laboratory-based x-ray phase-contrast tomography enables 3D virtual histology,” Proc. SPIE 9964, 99640I (2016).
[Crossref]

D. H. Larsson, W. Vågberg, A. Yaroshenko, A. Ö. Yildirim, and H. M. Hertz, “High-resolution short-exposure small-animal laboratory x-ray phase-contrast tomography,” Sci. Rep. 6, 39074 (2016).
[Crossref] [PubMed]

M. Lyubomirskiy, I. Snigireva, and A. Snigirev, “Lens coupled tunable Young’s double pinhole system for hard X-ray spatial coherence characterization,” Opt. Express 24, 13679–13686 (2016).
[Crossref] [PubMed]

2015 (2)

M. Krenkel, A. Markus, M. Bartels, C. Dullin, F. Alves, and T. Salditt, “Phase-contrast zoom tomography reveals precise locations of macrophages in mouse lungs,” Sci. Rep. 5, 09973 (2015).
[Crossref]

T. Salditt, M. Osterhoff, M. Krenkel, R. N. Wilke, M. Priebe, M. Bartels, S. Kalbfleisch, and M. Sprung, “Compound focusing mirror and X-ray waveguide optics for coherent imaging and nano-diffraction,” J. Synchrotron Rad. 22, 867–878 (2015).
[Crossref]

2014 (5)

J. Hagemann, A.-L. Robisch, D. R. Luke, C. Homann, T. Hohage, P. Cloetens, H. Suhonen, and T. Salditt, “Reconstruction of wave front and object for inline holography from a set of detection planes,” Opt. Express 22, 11552–11569 (2014).
[Crossref] [PubMed]

B. Enders, M. Dierolf, P. Cloetens, M. Stockmar, F. Pfeiffer, and P. Thibault, “Ptychography with broad-bandwidth radiation,” Appl. Phys. Lett. 104, 171104 (2014).
[Crossref]

P. M. Pelz, M. Guizar-Sicairos, P. Thibault, I. Johnson, M. Holler, and A. Menzel, “On-the-fly scans for X-ray ptychography,” Appl. Phys. Lett. 105, 251101 (2014).
[Crossref]

D. J. Batey, D. Claus, and J. M. Rodenburg, “Information multiplexing in ptychography,” Ultramicroscopy 138, 13–21 (2014).
[Crossref] [PubMed]

T. Mey, B. Schäfer, K. Mann, B. Keitel, M. Kuhlmann, and E. Plönjes, “Wigner distribution measurements of the spatial coherence properties of the free-electron laser FLASH,” Opt. Express 22, 16571–16584 (2014).
[Crossref] [PubMed]

2013 (3)

D. D. Mai, J. Hallmann, T. Reusch, M. Osterhoff, S. Düsterer, R. Treusch, A. Singer, M. Beckers, T. Gorniak, and T. Senkbeil, “Single pulse coherence measurements in the water window at the free-electron laser FLASH,” Opt. Express 21, 13005–13017 (2013).
[Crossref] [PubMed]

P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494, 68–71 (2013).
[Crossref] [PubMed]

M. Bartels, V. H. Hernandez, M. Krenkel, T. Moser, and T. Salditt, “Phase contrast tomography of the mouse cochlea at microfocus x-ray sources,” Appl. Phys. Lett. 103, 083703 (2013).
[Crossref]

2011 (4)

D. Pelliccia, A. Y. Nikulin, H. O. Moser, and K. A. Nugent, “Experimental characterization of the coherence properties of hard x-ray sources,” Opt. Express 19, 8073–8078 (2011).
[Crossref] [PubMed]

T. Salditt, S. Kalbfleisch, M. Osterhoff, S. P. Krüger, M. Bartels, K. Giewekemeyer, H. Neubauer, and M. Sprung, “Partially coherent nano-focused x-ray radiation characterized by Talbot interferometry,” Opt. Express 19, 9656–9675 (2011).
[Crossref] [PubMed]

K. A. Nugent, “The measurement of phase through the propagation of intensity: an introduction,” Contemp. Phys. 52, 55–69 (2011).
[Crossref]

B. Abbey, L. W. Whitehead, H. M. Quiney, D. J. Vine, G. A. Cadenazzi, C. A. Henderson, K. A. Nugent, E. Balaur, C. T. Putkunz, A. G. Peele, G. J. Williams, and I. Williams, “Lensless imaging using broadband X-ray sources,” Nat. Photonics 5, 420–424 (2011).
[Crossref]

2010 (2)

C. M. Kewish, M. Guizar-Sicairos, C. Liu, J. Qian, B. Shi, C. Benson, A. M. Khounsary, J. Vila-Comamala, O. Bunk, J. R. Fienup, A. T. Macrander, and L. Assoufid, “Reconstruction of an astigmatic hard X-ray beam and alignment of K-B mirrors from ptychographic coherent diffraction data,” Opt. Express 18, 23420–23427 (2010).
[Crossref] [PubMed]

A. Schropp, P. Boye, J. M. Feldkamp, R. Hoppe, J. Patommel, D. Samberg, S. Stephan, K. Giewekemeyer, R. N. Wilke, T. Salditt, J. Gulden, A. P. Mancuso, I. A. Vartanyants, E. Weckert, S. Schoder, M. Burghammer, and C. G. Schroer, “Hard x-ray nanobeam characterization by coherent diffraction microscopy,” Appl. Phys. Lett. 96, 091102 (2010).
[Crossref]

2009 (2)

S. Flewett, H. M. Quiney, C. Q. Tran, and K. A. Nugent, “Extracting coherent modes from partially coherent wavefields,” Opt. Lett. 34, 2198–2200 (2009).
[Crossref] [PubMed]

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[Crossref] [PubMed]

2008 (3)

C. T. Koch, “A flux-preserving non-linear inline holography reconstruction algorithm for partially coherent electrons,” Ultramicroscopy 108, 141–150 (2008).
[Crossref]

A. Singer, I. A. Vartanyants, M. Kuhlmann, S. Duesterer, R. Treusch, and J. Feldhaus, “Transverse-Coherence Properties of the Free-Electron-Laser FLASH at DESY,” Phys. Rev. Lett. 101, 254801 (2008).
[Crossref] [PubMed]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning X-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref] [PubMed]

2005 (2)

2002 (1)

D. R. Luke, J. V. Burke, and R. G. Lyon, “Optical wavefront reconstruction: Theory and numerical methods,” SIAM Rev. 44, 169–224 (2002).
[Crossref]

2001 (1)

L. Allen and M. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001).
[Crossref]

1999 (2)

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75, 2912–2914 (1999).
[Crossref]

J. Cheng and S. Han, “Phase imaging with partially coherent x rays,” Opt. Lett. 24, 175–177 (1999).
[Crossref]

1957 (1)

1949 (1)

J. V. Neumann, “On Rings of Operators. Reduction Theory,” Ann. Math. 50, 401–485 (1949).
[Crossref]

Abbey, B.

B. Abbey, L. W. Whitehead, H. M. Quiney, D. J. Vine, G. A. Cadenazzi, C. A. Henderson, K. A. Nugent, E. Balaur, C. T. Putkunz, A. G. Peele, G. J. Williams, and I. Williams, “Lensless imaging using broadband X-ray sources,” Nat. Photonics 5, 420–424 (2011).
[Crossref]

Allen, L.

L. Allen and M. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001).
[Crossref]

Alves, F.

M. Krenkel, A. Markus, M. Bartels, C. Dullin, F. Alves, and T. Salditt, “Phase-contrast zoom tomography reveals precise locations of macrophages in mouse lungs,” Sci. Rep. 5, 09973 (2015).
[Crossref]

Assoufid, L.

Balaur, E.

B. Abbey, L. W. Whitehead, H. M. Quiney, D. J. Vine, G. A. Cadenazzi, C. A. Henderson, K. A. Nugent, E. Balaur, C. T. Putkunz, A. G. Peele, G. J. Williams, and I. Williams, “Lensless imaging using broadband X-ray sources,” Nat. Photonics 5, 420–424 (2011).
[Crossref]

Bartels, M.

M. Krenkel, A. Markus, M. Bartels, C. Dullin, F. Alves, and T. Salditt, “Phase-contrast zoom tomography reveals precise locations of macrophages in mouse lungs,” Sci. Rep. 5, 09973 (2015).
[Crossref]

T. Salditt, M. Osterhoff, M. Krenkel, R. N. Wilke, M. Priebe, M. Bartels, S. Kalbfleisch, and M. Sprung, “Compound focusing mirror and X-ray waveguide optics for coherent imaging and nano-diffraction,” J. Synchrotron Rad. 22, 867–878 (2015).
[Crossref]

M. Bartels, V. H. Hernandez, M. Krenkel, T. Moser, and T. Salditt, “Phase contrast tomography of the mouse cochlea at microfocus x-ray sources,” Appl. Phys. Lett. 103, 083703 (2013).
[Crossref]

T. Salditt, S. Kalbfleisch, M. Osterhoff, S. P. Krüger, M. Bartels, K. Giewekemeyer, H. Neubauer, and M. Sprung, “Partially coherent nano-focused x-ray radiation characterized by Talbot interferometry,” Opt. Express 19, 9656–9675 (2011).
[Crossref] [PubMed]

Baruchel, J.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75, 2912–2914 (1999).
[Crossref]

Batey, D.

Batey, D. J.

D. J. Batey, D. Claus, and J. M. Rodenburg, “Information multiplexing in ptychography,” Ultramicroscopy 138, 13–21 (2014).
[Crossref] [PubMed]

Beckers, M.

Benson, C.

Boye, P.

A. Schropp, P. Boye, J. M. Feldkamp, R. Hoppe, J. Patommel, D. Samberg, S. Stephan, K. Giewekemeyer, R. N. Wilke, T. Salditt, J. Gulden, A. P. Mancuso, I. A. Vartanyants, E. Weckert, S. Schoder, M. Burghammer, and C. G. Schroer, “Hard x-ray nanobeam characterization by coherent diffraction microscopy,” Appl. Phys. Lett. 96, 091102 (2010).
[Crossref]

Bunk, O.

Burghammer, M.

A. Schropp, P. Boye, J. M. Feldkamp, R. Hoppe, J. Patommel, D. Samberg, S. Stephan, K. Giewekemeyer, R. N. Wilke, T. Salditt, J. Gulden, A. P. Mancuso, I. A. Vartanyants, E. Weckert, S. Schoder, M. Burghammer, and C. G. Schroer, “Hard x-ray nanobeam characterization by coherent diffraction microscopy,” Appl. Phys. Lett. 96, 091102 (2010).
[Crossref]

Burke, J. V.

D. R. Luke, J. V. Burke, and R. G. Lyon, “Optical wavefront reconstruction: Theory and numerical methods,” SIAM Rev. 44, 169–224 (2002).
[Crossref]

Cadenazzi, G. A.

B. Abbey, L. W. Whitehead, H. M. Quiney, D. J. Vine, G. A. Cadenazzi, C. A. Henderson, K. A. Nugent, E. Balaur, C. T. Putkunz, A. G. Peele, G. J. Williams, and I. Williams, “Lensless imaging using broadband X-ray sources,” Nat. Photonics 5, 420–424 (2011).
[Crossref]

Cheng, J.

Claus, D.

D. J. Batey, D. Claus, and J. M. Rodenburg, “Information multiplexing in ptychography,” Ultramicroscopy 138, 13–21 (2014).
[Crossref] [PubMed]

Cloetens, P.

B. Enders, M. Dierolf, P. Cloetens, M. Stockmar, F. Pfeiffer, and P. Thibault, “Ptychography with broad-bandwidth radiation,” Appl. Phys. Lett. 104, 171104 (2014).
[Crossref]

J. Hagemann, A.-L. Robisch, D. R. Luke, C. Homann, T. Hohage, P. Cloetens, H. Suhonen, and T. Salditt, “Reconstruction of wave front and object for inline holography from a set of detection planes,” Opt. Express 22, 11552–11569 (2014).
[Crossref] [PubMed]

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75, 2912–2914 (1999).
[Crossref]

David, C.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning X-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref] [PubMed]

Dierolf, M.

B. Enders, M. Dierolf, P. Cloetens, M. Stockmar, F. Pfeiffer, and P. Thibault, “Ptychography with broad-bandwidth radiation,” Appl. Phys. Lett. 104, 171104 (2014).
[Crossref]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning X-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref] [PubMed]

Duesterer, S.

A. Singer, I. A. Vartanyants, M. Kuhlmann, S. Duesterer, R. Treusch, and J. Feldhaus, “Transverse-Coherence Properties of the Free-Electron-Laser FLASH at DESY,” Phys. Rev. Lett. 101, 254801 (2008).
[Crossref] [PubMed]

Dullin, C.

M. Krenkel, A. Markus, M. Bartels, C. Dullin, F. Alves, and T. Salditt, “Phase-contrast zoom tomography reveals precise locations of macrophages in mouse lungs,” Sci. Rep. 5, 09973 (2015).
[Crossref]

Düsterer, S.

Edo, T.

Enders, B.

B. Enders, M. Dierolf, P. Cloetens, M. Stockmar, F. Pfeiffer, and P. Thibault, “Ptychography with broad-bandwidth radiation,” Appl. Phys. Lett. 104, 171104 (2014).
[Crossref]

Feldhaus, J.

A. Singer, I. A. Vartanyants, M. Kuhlmann, S. Duesterer, R. Treusch, and J. Feldhaus, “Transverse-Coherence Properties of the Free-Electron-Laser FLASH at DESY,” Phys. Rev. Lett. 101, 254801 (2008).
[Crossref] [PubMed]

Feldkamp, J. M.

A. Schropp, P. Boye, J. M. Feldkamp, R. Hoppe, J. Patommel, D. Samberg, S. Stephan, K. Giewekemeyer, R. N. Wilke, T. Salditt, J. Gulden, A. P. Mancuso, I. A. Vartanyants, E. Weckert, S. Schoder, M. Burghammer, and C. G. Schroer, “Hard x-ray nanobeam characterization by coherent diffraction microscopy,” Appl. Phys. Lett. 96, 091102 (2010).
[Crossref]

Fienup, J. R.

Flannery, B. P.

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M. Krenkel, A. Markus, M. Bartels, C. Dullin, F. Alves, and T. Salditt, “Phase-contrast zoom tomography reveals precise locations of macrophages in mouse lungs,” Sci. Rep. 5, 09973 (2015).
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Menzel, A.

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Moser, H. O.

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M. Bartels, V. H. Hernandez, M. Krenkel, T. Moser, and T. Salditt, “Phase contrast tomography of the mouse cochlea at microfocus x-ray sources,” Appl. Phys. Lett. 103, 083703 (2013).
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T. Salditt, M. Osterhoff, M. Krenkel, R. N. Wilke, M. Priebe, M. Bartels, S. Kalbfleisch, and M. Sprung, “Compound focusing mirror and X-ray waveguide optics for coherent imaging and nano-diffraction,” J. Synchrotron Rad. 22, 867–878 (2015).
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B. Abbey, L. W. Whitehead, H. M. Quiney, D. J. Vine, G. A. Cadenazzi, C. A. Henderson, K. A. Nugent, E. Balaur, C. T. Putkunz, A. G. Peele, G. J. Williams, and I. Williams, “Lensless imaging using broadband X-ray sources,” Nat. Photonics 5, 420–424 (2011).
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Pelz, P. M.

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B. Enders, M. Dierolf, P. Cloetens, M. Stockmar, F. Pfeiffer, and P. Thibault, “Ptychography with broad-bandwidth radiation,” Appl. Phys. Lett. 104, 171104 (2014).
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T. Salditt, M. Osterhoff, M. Krenkel, R. N. Wilke, M. Priebe, M. Bartels, S. Kalbfleisch, and M. Sprung, “Compound focusing mirror and X-ray waveguide optics for coherent imaging and nano-diffraction,” J. Synchrotron Rad. 22, 867–878 (2015).
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M. Töpperwien, M. Krenkel, F. Quade, and T. Salditt, “Laboratory-based x-ray phase-contrast tomography enables 3D virtual histology,” Proc. SPIE 9964, 99640I (2016).
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B. Abbey, L. W. Whitehead, H. M. Quiney, D. J. Vine, G. A. Cadenazzi, C. A. Henderson, K. A. Nugent, E. Balaur, C. T. Putkunz, A. G. Peele, G. J. Williams, and I. Williams, “Lensless imaging using broadband X-ray sources,” Nat. Photonics 5, 420–424 (2011).
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Roberts, A.

Robisch, A.-L.

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Rodenburg, J. M.

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Salditt, T.

J. Hagemann, A.-L. Robisch, M. Osterhoff, and T. Salditt, “Probe reconstruction for holographic X-ray imaging,” J. Synchrotron Rad. 24, 498–505 (2017).
[Crossref]

M. Töpperwien, M. Krenkel, F. Quade, and T. Salditt, “Laboratory-based x-ray phase-contrast tomography enables 3D virtual histology,” Proc. SPIE 9964, 99640I (2016).
[Crossref]

M. Krenkel, A. Markus, M. Bartels, C. Dullin, F. Alves, and T. Salditt, “Phase-contrast zoom tomography reveals precise locations of macrophages in mouse lungs,” Sci. Rep. 5, 09973 (2015).
[Crossref]

T. Salditt, M. Osterhoff, M. Krenkel, R. N. Wilke, M. Priebe, M. Bartels, S. Kalbfleisch, and M. Sprung, “Compound focusing mirror and X-ray waveguide optics for coherent imaging and nano-diffraction,” J. Synchrotron Rad. 22, 867–878 (2015).
[Crossref]

J. Hagemann, A.-L. Robisch, D. R. Luke, C. Homann, T. Hohage, P. Cloetens, H. Suhonen, and T. Salditt, “Reconstruction of wave front and object for inline holography from a set of detection planes,” Opt. Express 22, 11552–11569 (2014).
[Crossref] [PubMed]

M. Bartels, V. H. Hernandez, M. Krenkel, T. Moser, and T. Salditt, “Phase contrast tomography of the mouse cochlea at microfocus x-ray sources,” Appl. Phys. Lett. 103, 083703 (2013).
[Crossref]

T. Salditt, S. Kalbfleisch, M. Osterhoff, S. P. Krüger, M. Bartels, K. Giewekemeyer, H. Neubauer, and M. Sprung, “Partially coherent nano-focused x-ray radiation characterized by Talbot interferometry,” Opt. Express 19, 9656–9675 (2011).
[Crossref] [PubMed]

A. Schropp, P. Boye, J. M. Feldkamp, R. Hoppe, J. Patommel, D. Samberg, S. Stephan, K. Giewekemeyer, R. N. Wilke, T. Salditt, J. Gulden, A. P. Mancuso, I. A. Vartanyants, E. Weckert, S. Schoder, M. Burghammer, and C. G. Schroer, “Hard x-ray nanobeam characterization by coherent diffraction microscopy,” Appl. Phys. Lett. 96, 091102 (2010).
[Crossref]

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A. Schropp, P. Boye, J. M. Feldkamp, R. Hoppe, J. Patommel, D. Samberg, S. Stephan, K. Giewekemeyer, R. N. Wilke, T. Salditt, J. Gulden, A. P. Mancuso, I. A. Vartanyants, E. Weckert, S. Schoder, M. Burghammer, and C. G. Schroer, “Hard x-ray nanobeam characterization by coherent diffraction microscopy,” Appl. Phys. Lett. 96, 091102 (2010).
[Crossref]

Schäfer, B.

Schlenker, M.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75, 2912–2914 (1999).
[Crossref]

Schoder, S.

A. Schropp, P. Boye, J. M. Feldkamp, R. Hoppe, J. Patommel, D. Samberg, S. Stephan, K. Giewekemeyer, R. N. Wilke, T. Salditt, J. Gulden, A. P. Mancuso, I. A. Vartanyants, E. Weckert, S. Schoder, M. Burghammer, and C. G. Schroer, “Hard x-ray nanobeam characterization by coherent diffraction microscopy,” Appl. Phys. Lett. 96, 091102 (2010).
[Crossref]

Schroer, C. G.

A. Schropp, P. Boye, J. M. Feldkamp, R. Hoppe, J. Patommel, D. Samberg, S. Stephan, K. Giewekemeyer, R. N. Wilke, T. Salditt, J. Gulden, A. P. Mancuso, I. A. Vartanyants, E. Weckert, S. Schoder, M. Burghammer, and C. G. Schroer, “Hard x-ray nanobeam characterization by coherent diffraction microscopy,” Appl. Phys. Lett. 96, 091102 (2010).
[Crossref]

Schropp, A.

A. Schropp, P. Boye, J. M. Feldkamp, R. Hoppe, J. Patommel, D. Samberg, S. Stephan, K. Giewekemeyer, R. N. Wilke, T. Salditt, J. Gulden, A. P. Mancuso, I. A. Vartanyants, E. Weckert, S. Schoder, M. Burghammer, and C. G. Schroer, “Hard x-ray nanobeam characterization by coherent diffraction microscopy,” Appl. Phys. Lett. 96, 091102 (2010).
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Senkbeil, T.

Shi, B.

Singer, A.

D. D. Mai, J. Hallmann, T. Reusch, M. Osterhoff, S. Düsterer, R. Treusch, A. Singer, M. Beckers, T. Gorniak, and T. Senkbeil, “Single pulse coherence measurements in the water window at the free-electron laser FLASH,” Opt. Express 21, 13005–13017 (2013).
[Crossref] [PubMed]

A. Singer, I. A. Vartanyants, M. Kuhlmann, S. Duesterer, R. Treusch, and J. Feldhaus, “Transverse-Coherence Properties of the Free-Electron-Laser FLASH at DESY,” Phys. Rev. Lett. 101, 254801 (2008).
[Crossref] [PubMed]

Snigirev, A.

Snigireva, I.

Sprung, M.

T. Salditt, M. Osterhoff, M. Krenkel, R. N. Wilke, M. Priebe, M. Bartels, S. Kalbfleisch, and M. Sprung, “Compound focusing mirror and X-ray waveguide optics for coherent imaging and nano-diffraction,” J. Synchrotron Rad. 22, 867–878 (2015).
[Crossref]

T. Salditt, S. Kalbfleisch, M. Osterhoff, S. P. Krüger, M. Bartels, K. Giewekemeyer, H. Neubauer, and M. Sprung, “Partially coherent nano-focused x-ray radiation characterized by Talbot interferometry,” Opt. Express 19, 9656–9675 (2011).
[Crossref] [PubMed]

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A. Schropp, P. Boye, J. M. Feldkamp, R. Hoppe, J. Patommel, D. Samberg, S. Stephan, K. Giewekemeyer, R. N. Wilke, T. Salditt, J. Gulden, A. P. Mancuso, I. A. Vartanyants, E. Weckert, S. Schoder, M. Burghammer, and C. G. Schroer, “Hard x-ray nanobeam characterization by coherent diffraction microscopy,” Appl. Phys. Lett. 96, 091102 (2010).
[Crossref]

Stockmar, M.

B. Enders, M. Dierolf, P. Cloetens, M. Stockmar, F. Pfeiffer, and P. Thibault, “Ptychography with broad-bandwidth radiation,” Appl. Phys. Lett. 104, 171104 (2014).
[Crossref]

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Teukolsky, S. A.

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Thibault, P.

B. Enders, M. Dierolf, P. Cloetens, M. Stockmar, F. Pfeiffer, and P. Thibault, “Ptychography with broad-bandwidth radiation,” Appl. Phys. Lett. 104, 171104 (2014).
[Crossref]

P. M. Pelz, M. Guizar-Sicairos, P. Thibault, I. Johnson, M. Holler, and A. Menzel, “On-the-fly scans for X-ray ptychography,” Appl. Phys. Lett. 105, 251101 (2014).
[Crossref]

P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494, 68–71 (2013).
[Crossref] [PubMed]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning X-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref] [PubMed]

Thompson, B. J.

Töpperwien, M.

M. Töpperwien, M. Krenkel, F. Quade, and T. Salditt, “Laboratory-based x-ray phase-contrast tomography enables 3D virtual histology,” Proc. SPIE 9964, 99640I (2016).
[Crossref]

Tran, C. Q.

Treusch, R.

D. D. Mai, J. Hallmann, T. Reusch, M. Osterhoff, S. Düsterer, R. Treusch, A. Singer, M. Beckers, T. Gorniak, and T. Senkbeil, “Single pulse coherence measurements in the water window at the free-electron laser FLASH,” Opt. Express 21, 13005–13017 (2013).
[Crossref] [PubMed]

A. Singer, I. A. Vartanyants, M. Kuhlmann, S. Duesterer, R. Treusch, and J. Feldhaus, “Transverse-Coherence Properties of the Free-Electron-Laser FLASH at DESY,” Phys. Rev. Lett. 101, 254801 (2008).
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Supplementary Material (1)

NameDescription
» Code 1       Algorithm and simulation scripts for near-field coherent mode reconstruction

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Figures (10)

Fig. 1
Fig. 1 Sketch of the data generation for the numerical experiment. (left) 3 exemplary modes, which are individually propagated to the detection planes at Fresnel numbers Fr k (right). (bottom) The corresponding k is formed by incoherently adding the individual intensities Ik,m.
Fig. 2
Fig. 2 Generation of coherent modes. Test images (a) were taken and interpreted as amplitudes and phases of the coherent modes. Using these modes as input for a QR-factorization, suitable, i.e. orthogonal, modes (b) are obtained for propagation and data generation, following the scheme introduced in Fig. 1. The modes are scaled according to their occupation number. (7)
Fig. 3
Fig. 3 Results of the mmMMP reconstruction. (a) The set of reconstructed modes in phase and amplitude. (b) The convergence graph for the occupation numbers λm. The convergence is shown relatively to the current occupation in an iteration with respect to the prescribed occupation λ m 0 . (c) Comparison of the error evolution Δ k of the mean over all measurement planes Eq. (12) for a single mode reconstruction (red) and multi mode reconstruction (blue). Dashed lines indicate the bounds of the standard deviation over all measurement planes.
Fig. 4
Fig. 4 Comparison of the degree of coherence ∥j(d)∥ for the reconstructed modes (red) and input modes (blue).
Fig. 5
Fig. 5 Example for a noisy measurement at Fr = 3.2−3. (top) The realization for µ = 10 photons per pixel and (bottom) µ = 1000 photons per pixel. The insert (left) shows the loss of feature visibility in the hologram by noise. Note, that for representation purposes, the intensities have been rescaled by their mean value. Scale bar indicates 50 px.
Fig. 6
Fig. 6 Results obtained from noisy data. (a) Reconstructed modes in phase and amplitude. The left half shows the reconstruction for µ = 10 and the right for µ = 1000. The amplitudes have been normalized by the mean value for the side-by-side plot. (b) Error Δ as a function of iteration n, for both photon numbers. Dashed lines indicate the bounds of the standard deviation over all measurement planes. (c) Reconstruction of Ψ3 for µ = 10 after 95000 iterations (left) compared to the input mode (right), cf. Fig. 2 of the main manuscript.
Fig. 7
Fig. 7 Results obtained from the reduced input data set. (a) Reconstructed modes in phase and amplitude after 50000 iterations. The left half shows the reconstruction for K = 6 and the right for K = 8. (b) Error evolution for both K as function of iteration number. Dashed lines indicate the bounds of the standard deviation over all measurement planes. (c) Reconstruction of Ψ3 for K = 8 after 270000 iterations (left) compared to the input mode (right), cf. Fig. 2.
Fig. 8
Fig. 8 Influence of the choice of M on Δ k . The results for M = 1 and M = 3 are already shown in Fig. 3(c). The results for M = 3 and M = 4 have been added. Solid lines indicate the mean error Δ k over all k . Dashed lines indicate the bounds of the standard deviation over all measurement planes.
Fig. 9
Fig. 9 Results of the equal occupation reconstruction. (a) The set of reconstructed modes in phase and amplitude. (b) The convergence graph for the occupation numbers λm. The convergence is shown relatively to the current occupation in an iteration with respect to the prescribed occupation λ m 0 . (c) Comparison of the error evolution Δ k of the mean over all measurement planes Eq. (12) for a single mode reconstruction (red) and multi mode reconstruction (blue). Dashed lines indicate the bounds of the standard deviation over all measurement planes.
Fig. 10
Fig. 10 Trajectories for the realizations of λm. The individual trajectories (a) of the relative occupation λ m / λ m 0 for overall 12 realizations are shown, using the parameters listed in Tab. 1. The realization with λ m ^ = { 0.487 , 0.291 , 0.222 } showing not satisfactory convergence after 30000 iterations, has been iterated 70000 iterations more (b).

Tables (1)

Tables Icon

Table 1 Summary of the parameters for the numerical experiment.

Equations (12)

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Ψ m * , Ψ n = δ n m ,
I m = | D Fr ( Ψ m ) | 2 ,
= m λ m I m ,
J m ( r 1 , r 2 ) = Ψ m * ( r 1 ) Ψ m ( r 2 ) .
j ( r 1 , r 2 ) = J ( r 1 , r 2 ) J ( r 1 , r 1 ) ( r 2 , r 2 ) .
Ψ n + 1 = β n 2 ( R S ( R ( Ψ n ) ) + Ψ n ) + ( 1 β n ) P ( Ψ n ) ,
P k ( Ψ m ) D Fr k ( A k [ D Fr k ( Ψ m ) ] ) ,
A k ( ) = I k , m m = 1 M I k , m k exp ( i arg ( ) ) ,
D Fr k ( ) = 1 [ [ ] exp ( ( i π ) / ( 2 Fr k ) ( k x 2 + k y 2 ) ) ] ,
P ( Ψ n ) { 1 K k = 1 K P k ( Ψ 1 ) 1 K k = 1 K P k ( Ψ M ) .
P S ( Ψ n ) = Q R ( Ψ n ) .
Δ k = all pixels | m I k , m k | 2 / N .

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